Are both of these indefinite integrals?Scenario 1: (1) f ( x ) := ∫ cos ⁡ ( x ) d x Scenario 2: (2) g ( x ) := ∫ a x cos ⁡ ( t ) d t [ edited after seeing @hardmath's comment ] [ a is a constant ] Comments:Both (1) and (2) are functions. An antiderivative/indefinite integral is a function, while a definite integral is a number. So, (3) f ( x ) = g ( x ) = sin ⁡ ( x ) + c [ c is a constant ] Is line (3) correct? Are both (1) and (2) antiderivatives/indefinite integrals?

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Are both of these indefinite integrals?Scenario 1:                     (1)                    f        (        x        )        :=        ∫                  cos          ⁡          (          x          )          d          x                    Scenario 2:                     (2)                    g        (        x        )        :=                  ∫                      a                                x                                    cos          ⁡          (          t          )          d          t                        [        edited after seeing @hardmath&#039;s comment        ]                [        a is a constant        ]            Comments:Both (1) and (2) are functions. An antiderivative/indefinite integral is a function, while a definite integral is a number. So,                     (3)                    f        (        x        )        =        g        (        x        )        =        sin        ⁡        (        x        )        +        c                [        c is a constant        ]            Is line (3) correct? Are both (1) and (2) antiderivatives/indefinite integrals?

Cristal Obrien · Accepted Answer

Step 1Not quite. (1) is definitely (no pun intended) an indefinite integral. Note that evaluating an indefinite integral does not actually lead to a function, but rather to a family of functions differing by a constant. In the case of (1), we have   f  (  x  )  =  sin  ⁡  x  +  C (a family of functions).Step 2(2) is simply a function for any given a: by the Fundamental Theorem of Calculus,   g  (  x  )  =  sin  ⁡  x  −  sin  ⁡  a. Note that this really is one function, not a family of functions. (If one takes a family of parameters a, then that would result in a family of functions   g  (  x  )  =      g    a    (  x  ); but it still wouldn&#039;t result in every possible function in the family from (1), since sin a is bounded.)

Are both of these indefinite integrals? Scenario 1: <mtable displaystyle="true"> <mlabeled

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